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Selfdual codes and invariant theory
 MATH. NACHRICHTEN
, 2006
"... There is a beautiful analogy between most of the notions for lattices and codes and it seems to be quite promising to develop coding theory analogues of concepts known in the theory of lattices and modular forms and vice versa. Some of these analogies are presented in this short note that intends to ..."
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There is a beautiful analogy between most of the notions for lattices and codes and it seems to be quite promising to develop coding theory analogues of concepts known in the theory of lattices and modular forms and vice versa. Some of these analogies are presented in this short note that intends to survey recent developments connected to my talk HeckeOperators for codes in Luminy, on May 9, 2007, where I introduce the KneserHeckeOperators mentioned in Section 3.5. More details can be found in the paper [7], a preprint of which is available on my homepage.
An analogue of Heckeoperators in coding theory.
, 2005
"... The KneserHeckeoperator is a linear operator defined on the complex vector space spanned by the equivalence classes of a family of selfdual codes of fixed length. It maps a linear selfdual code C over a finite field to the formal sum of the equivalence classes of those selfdual codes that inter ..."
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The KneserHeckeoperator is a linear operator defined on the complex vector space spanned by the equivalence classes of a family of selfdual codes of fixed length. It maps a linear selfdual code C over a finite field to the formal sum of the equivalence classes of those selfdual codes that intersect C in a codimension 1 subspace. The eigenspaces of this selfadjoint linear operator may be described in terms of a codingtheory analogue of the Siegel Φoperator. MSC: 94B05, 11F60
Finite Weilrepresentations and associated Heckealgebras
, 2006
"... 1 An algebra H(Gm) of doublecosets is constructed for every finite Weilrepresentation Gm. For the CliffordWeil groups Gm = Cm(ρ) associated to some classical Type ρ of selfdual codes over a finite field, this algebra is shown to be commutative. Then the eigenspacedecomposition of H(Cm(ρ)) acting ..."
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1 An algebra H(Gm) of doublecosets is constructed for every finite Weilrepresentation Gm. For the CliffordWeil groups Gm = Cm(ρ) associated to some classical Type ρ of selfdual codes over a finite field, this algebra is shown to be commutative. Then the eigenspacedecomposition of H(Cm(ρ)) acting on the space of degree N invariants of Cm(ρ) may be obtained from the kernels of powers of the coding theory analogue of the Siegel Φoperator. 1 Introduction. The present paper continues the investigation of the parallels between lattices and codes in particular those analogies that are reflected in the theory of modular forms and invariant theory of certain finite groups. Degreem Siegel theta series of lattices are modular forms for certain subgroups of Sp2m(R). Similarly degreem complete weightenumerators
Note on the Biweight Enumerators of SelfDual Codes over Z_k
, 1999
"... Recently there has been interest in selfdual codes over finite rings. In this note, gfold joint weight enumerators and gfold multiweight enumerators of codes over the ring Z k of integers modulo k are introduced as a generalization of the biweight enumerators. We investigate these weight enumer ..."
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Recently there has been interest in selfdual codes over finite rings. In this note, gfold joint weight enumerators and gfold multiweight enumerators of codes over the ring Z k of integers modulo k are introduced as a generalization of the biweight enumerators. We investigate these weight enumerators and the biweight enumerators of selfdual codes over Z k . The biweight enumerator of a class of binary codes, introduced in this note, is also studied. We derive Gleasontype theorems for the corresponding biweight enumerators with the help of invariant theory. 1 Introduction The conditions satisfied by the biweight enumerators of binary Type I codes and Type II codes were studied in [7] and [5], respectively. Using invariant theory, a basis for the space of invariants, which the biweight enumerator for such codes belongs, was also given. In this note, gfold joint weight enumerators and gfold multiweight enumerators of codes over the ring Z k of integers modulo k are introduced ...